Immersing n-spheres in 2n-space (Ex)

Latest revision as of 14:49, 1 April 2012

Let $<wikitex>; Let $V_{n, k}$ denote the Stiefel manifold of orthogonal $k$-frames in $\R^n$ and consider the following fibration sequence $$V_{n, n} \to V_{2n, 2n} \to V_{2n, n}.$$ Complete the following: * Use the [[Hirsch-Smale immersion theorem]] to give an interpretation of the boundary map in the homotopy exact sequence of the above fibration: $$ \partial \colon \pi_n(V_{2n, n}) \to \pi_{n-1}(V_{n, n}) $$ * Hence prove the following: if $n \neq 1, 3, 7$, any immersion $f \colon S^n \to \Rr^{2n}$ is regularly homotopic to an embedding if and only if the normal bundle of $f$ is trivial. * Given an example of an immersion $S^1 \to \Rr^2$ with trivial normal bundle and which is not regularly homotopic to an embedding. </wikitex> == References == {{#RefList:}} [[Category:Exercises]]V_{n, k}$ denote the Stiefel manifold of orthogonal $k$-frames in $\R^n$ and consider the following fibration sequence

$$\displaystyle V_{n, n} \to V_{2n, 2n} \to V_{2n, n}.$$

Complete the following:

• Use the Hirsch-Smale immersion theorem to give an interpretation of the boundary map in the homotopy exact sequence of the above fibration:

$$\displaystyle \partial \colon \pi_n(V_{2n, n}) \to \pi_{n-1}(V_{n, n})$$

• Hence prove the following: if $n \neq 1, 3, 7$, any immersion $f \colon S^n \to \Rr^{2n}$ is regularly homotopic to an embedding if and only if the normal bundle of $f$ is trivial.
• Given an example of an immersion $S^1 \to \Rr^2$ with trivial normal bundle and which is not regularly homotopic to an embedding.